| In this article, we study the monotonicity and completely monotonic for the functions which is involving the Gamma function, Psi function and Polygamma functions, and furtherly we consider logarithmically completely monotonicity properties of functions which are constructed of several types correlation functions. At the same time, relevant inequalities for these functions have been proved.The content of this paper can be divided into three parts:1. The theory of completely monotonic:First we discussed monotonic properties of function that is difference of two completely monotonic function, then we simply promotion the coefficient of function, furtherly we get the necessary and sufficient condition for complete monotonic; Finally we research monotonic problem of function is related to Polygamma function.2. The theory of logarithmically completely monotonic:This paper first by introducing a new function proves that the existed ratio of Gamma function is strictly logarithmically completely monotonic; The author also proves that generalized the coefficient of variable of some ratios of the Gamma function is strictly logarithmically completely monotonic; At last, we base on the existed functions, some relevant functions are constructed, at the same time, we proves this new functions are logarithmically completely monotonic.3. The theory of inequalities:We have discussed the inequalities in great detail and generalized the latest results. We prove inequalities about Gamma function, Psi function and the ratio of Gamma function through Lagrange median theorem. Then we compare the new results with the old. |
